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YasuakiHonda/CFtoSerieseByEuler.ipynb

Created May 29, 2021 01:52
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CFtoSerieseByEuler.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{12}$}{\\it applyLFT}\\left(M , z\\right):=\\frac{M_{1,1}\\,z+M_{1,2}}{M_{2,1}\\,z+M_{2,2}}\\]"
],
"text/plain": [
" M z + M\n",
" 1, 1 1, 2\n",
"(%o12) applyLFT(M, z) := ---------------\n",
" M z + M\n",
" 2, 1 2, 2"
],
"text/x-maxima": [
"applyLFT(M,z):=(M[1,1]*z+M[1,2])/(M[2,1]*z+M[2,2])"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(M,z):=(M[1,1]*z+M[1,2])/(M[2,1]*z+M[2,2]);"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{14}$}{\\it genmat}\\left(n\\right):=\\begin{pmatrix}a_{n} & b_{n} \\\\ 1 & 0 \\\\ \\end{pmatrix}\\]"
],
"text/plain": [
" [ a b ]\n",
"(%o14) genmat(n) := [ n n ]\n",
" [ ]\n",
" [ 1 0 ]"
],
"text/x-maxima": [
"genmat(n):=matrix([a[n],b[n]],[1,0])"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"genmat(n):=matrix([a[n],b[n]],[1,0]);"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{23}$}\\frac{a_{1}\\,a_{2}+b_{1}}{a_{2}}\\]"
],
"text/plain": [
" a a + b\n",
" 1 2 1\n",
"(%o23) ----------\n",
" a\n",
" 2"
],
"text/x-maxima": [
"(a[1]*a[2]+b[1])/a[2]"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(genmat(1),a[2]);"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{24}$}\\frac{a_{1}\\,a_{2}\\,a_{3}+b_{1}\\,a_{3}+a_{1}\\,b_{2}}{a_{2}\\,a_{3}+b_{2}}\\]"
],
"text/plain": [
" a a a + b a + a b\n",
" 1 2 3 1 3 1 2\n",
"(%o24) ------------------------\n",
" a a + b\n",
" 2 3 2"
],
"text/x-maxima": [
"(a[1]*a[2]*a[3]+b[1]*a[3]+a[1]*b[2])/(a[2]*a[3]+b[2])"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(genmat(1).genmat(2),a[3]),factor;"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{27}$}\\frac{a_{1}\\,a_{2}\\,a_{3}\\,a_{4}+b_{1}\\,a_{3}\\,a_{4}+a_{1}\\,b_{2}\\,a_{4}+a_{1}\\,a_{2}\\,b_{3}+b_{1}\\,b_{3}}{a_{2}\\,a_{3}\\,a_{4}+b_{2}\\,a_{4}+a_{2}\\,b_{3}}\\]"
],
"text/plain": [
" a a a a + b a a + a b a + a a b + b b\n",
" 1 2 3 4 1 3 4 1 2 4 1 2 3 1 3\n",
"(%o27) ----------------------------------------------------\n",
" a a a + b a + a b\n",
" 2 3 4 2 4 2 3"
],
"text/x-maxima": [
"(a[1]*a[2]*a[3]*a[4]+b[1]*a[3]*a[4]+a[1]*b[2]*a[4]+a[1]*a[2]*b[3]+b[1]*b[3])\n",
" /(a[2]*a[3]*a[4]+b[2]*a[4]+a[2]*b[3])"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(genmat(1).genmat(2).genmat(3),a[4]),factor;"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{28}$}\\frac{a_{1}\\,a_{2}\\,a_{3}\\,a_{4}\\,a_{5}+b_{1}\\,a_{3}\\,a_{4}\\,a_{5}+a_{1}\\,b_{2}\\,a_{4}\\,a_{5}+a_{1}\\,a_{2}\\,b_{3}\\,a_{5}+b_{1}\\,b_{3}\\,a_{5}+a_{1}\\,a_{2}\\,a_{3}\\,b_{4}+b_{1}\\,a_{3}\\,b_{4}+a_{1}\\,b_{2}\\,b_{4}}{a_{2}\\,a_{3}\\,a_{4}\\,a_{5}+b_{2}\\,a_{4}\\,a_{5}+a_{2}\\,b_{3}\\,a_{5}+a_{2}\\,a_{3}\\,b_{4}+b_{2}\\,b_{4}}\\]"
],
"text/plain": [
"(%o28) (a a a a a + b a a a + a b a a + a a b a + b b a\n",
" 1 2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 5 1 3 5\n",
" + a a a b + b a b + a b b )/(a a a a + b a a + a b a\n",
" 1 2 3 4 1 3 4 1 2 4 2 3 4 5 2 4 5 2 3 5\n",
" + a a b + b b )\n",
" 2 3 4 2 4"
],
"text/x-maxima": [
"(a[1]*a[2]*a[3]*a[4]*a[5]+b[1]*a[3]*a[4]*a[5]+a[1]*b[2]*a[4]*a[5]\n",
" +a[1]*a[2]*b[3]*a[5]+b[1]*b[3]*a[5]\n",
" +a[1]*a[2]*a[3]*b[4]+b[1]*a[3]*b[4]+a[1]*b[2]*b[4])\n",
" /(a[2]*a[3]*a[4]*a[5]+b[2]*a[4]*a[5]+a[2]*b[3]*a[5]+a[2]*a[3]*b[4]+b[2]*b[4])"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(genmat(1).genmat(2).genmat(3).genmat(4),a[5]),factor;"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{29}$}{\\it mprod}\\left(n\\right):=\\mathbf{if}\\;n=1\\;\\mathbf{then}\\;{\\it genmat}\\left(1\\right)\\;\\mathbf{else}\\;{\\it mprod}\\left(n-1\\right)\\cdot {\\it genmat}\\left(n\\right)\\]"
],
"text/plain": [
"(%o29) mprod(n) := if n = 1 then genmat(1) else mprod(n - 1) . genmat(n)"
],
"text/x-maxima": [
"mprod(n):=if n = 1 then genmat(1) else mprod(n-1) . genmat(n)"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mprod(n):=if n=1 then genmat(1) else mprod(n-1).genmat(n);"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{37}$}\\frac{b_{1}}{a_{2}}\\]"
],
"text/plain": [
" b\n",
" 1\n",
"(%o37) --\n",
" a\n",
" 2"
],
"text/x-maxima": [
"b[1]/a[2]"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(mprod(1),a[2])-a[1],factor;"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{36}$}-\\frac{b_{1}\\,b_{2}}{a_{2}\\,\\left(a_{2}\\,a_{3}+b_{2}\\right)}\\]"
],
"text/plain": [
" b b\n",
" 1 2\n",
"(%o36) - ---------------\n",
" a (a a + b )\n",
" 2 2 3 2"
],
"text/x-maxima": [
"-(b[1]*b[2])/(a[2]*(a[2]*a[3]+b[2]))"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(mprod(2),a[3])-applyLFT(mprod(1),a[2]),factor;"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{35}$}\\frac{b_{1}\\,b_{2}\\,b_{3}}{\\left(a_{2}\\,a_{3}+b_{2}\\right)\\,\\left(a_{2}\\,a_{3}\\,a_{4}+b_{2}\\,a_{4}+a_{2}\\,b_{3}\\right)}\\]"
],
"text/plain": [
" b b b\n",
" 1 2 3\n",
"(%o35) ---------------------------------------\n",
" (a a + b ) (a a a + b a + a b )\n",
" 2 3 2 2 3 4 2 4 2 3"
],
"text/x-maxima": [
"(b[1]*b[2]*b[3])/((a[2]*a[3]+b[2])*(a[2]*a[3]*a[4]+b[2]*a[4]+a[2]*b[3]))"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(mprod(3),a[4])-applyLFT(mprod(2),a[3]),factor;"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\[\\tag{${\\it \\%o}_{34}$}-\\frac{b_{1}\\,b_{2}\\,b_{3}\\,b_{4}}{\\left(a_{2}\\,a_{3}\\,a_{4}+b_{2}\\,a_{4}+a_{2}\\,b_{3}\\right)\\,\\left(a_{2}\\,a_{3}\\,a_{4}\\,a_{5}+b_{2}\\,a_{4}\\,a_{5}+a_{2}\\,b_{3}\\,a_{5}+a_{2}\\,a_{3}\\,b_{4}+b_{2}\\,b_{4}\\right)}\\]"
],
"text/plain": [
"(%o34) - (b b b b )/((a a a + b a + a b )\n",
" 1 2 3 4 2 3 4 2 4 2 3\n",
" (a a a a + b a a + a b a + a a b + b b ))\n",
" 2 3 4 5 2 4 5 2 3 5 2 3 4 2 4"
],
"text/x-maxima": [
"-(b[1]*b[2]*b[3]*b[4])/((a[2]*a[3]*a[4]+b[2]*a[4]+a[2]*b[3])\n",
" *(a[2]*a[3]*a[4]*a[5]+b[2]*a[4]*a[5]+a[2]*b[3]*a[5]\n",
" +a[2]*a[3]*b[4]+b[2]*b[4]))"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"applyLFT(mprod(4),a[5])-applyLFT(mprod(3),a[4]),factor;"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
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"display_name": "Maxima",
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